How exactly does Adenosine Triphosphate (ATP) provide energy to other molecules? I know that energy is released when ATP becomes ADP because the phosphates groups are all negatively charged, therefore they repel each other and want to break away and be as far apart from each other as quickly as possible. However, what type of energy is released when that happens? Is it just the inorganic phophate that's released that carries all the energy? Or is it all lost to heat energy?
Also, what about the inorganic phosphate group that is released through hydrolysis. Does it simply bond to other molecules to excite them and thus make them do work? But how does the inorganic phophate bond to other molecules? Is it an ionic bond? And how exactly does it make them unstable?

Well to answer your first question, when ATP becomes ADP, some energy is lost as heat due to the second law of thermodynamics. Unfortunately I can not answer your second question but I'm sure some one else knows...

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-Albert Einstein

I don't know the exact mechanism. However, when the ATP molecule is hydrolyzed by an enzyme to ADP + Pi, the chemical potential energy in the broken bond is channelled by the enzyme to create other high potential energy bonds, to accelerate things (like in muscular movement), etc. Basically the energy goes directly from the ATP to an enzyme which then does whatever the enzyme does with the energy. The inorganic phosphate ideally doesn't have much kinetic energy at all. If it does, this is basically wasted energy that the enzyme should have captured and channelled.

Well, you all know that ATP is basically like having cash, it allows you to do "stuff." The thing that makes ATP a great energy source is because of its three phosphates. It isn't a stable molecule, which is why it can release a lot of energy in order to power muscle cells and do "work."

The thing about ATP is that it's made of three phosphate groups. The phosphate group itself has a negative three charge, so removing one phosphate group upon hydrolysis makes it ADP. A further hydrolysis of ADP doesn't provide nearly as much because it is far more stable than ATP. "Stable" molecules are just those that form spontaneously. Most of the time ATP is just made by something like an ATPase or proton gradient like in plants. You're also familiar with cellular respiration.

ATP is used because it is needed to phosphorylate things, so to answer your question the mechanism just involves kinases and phosphotases (which phosphorylate and de-phosphorylate respectively). In things like cell development where you have meiosis I and II, it's crucial to have ATP, and its mechanism in frogs with melanophores (pigment cells) can either aggregate or disperse when using kinases.

There is also glucose level controls with the pathway that is dysfunctional in diabetics. I'm not actually sure about the exact pathway, but needless to say kinases also play a central role in that too.

This is chemical energy-the energy of bond breaking and formation between atoms, not kinetic, and the unit is kJ/mole.

Is it just the inorganic phophate that's released that carries all the energy?

No, the inorganic phosphate doesn't carry the energy. The energy resides in the bond between the 2nd and 3rd phosphate groups.

Or is it all lost to heat energy?

Alot is, but not all. If all the energy went to heat then there would be no energy left for the enzymes to do its thing(s).

Also, what about the inorganic phosphate group that is released through hydrolysis. Does it simply bond to other molecules to excite them and thus make them do work?

Sometimes, but not always. Its reaction specific. Sometimes the phospate is bound to the substrate (phosphorylation) and sometimes the hydrolysis of the third phosphate group is used to provide energy for a reaction that requires it to go.

But how does the inorganic phophate bond to other molecules? Is it an ionic bond? And how exactly does it make them unstable?

Covalent bonding. Phospate group bonding is an inherently high negative free energy states. The hydrolysis of a phosphate groups is almost always an energy releasing reaction. For example, 31kJ/mol of energy is released when ATP becomes ADP. Another 31kJ/mol of energy is released with ADP becomes AMP, and 14kJ of energy is released with AMP becomes adenosine.